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## Division of Polynomials

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**Division by a Monomial**When dividing monomials with Exponential expressions, use the Quotient Rule of Exponents: if x = 0**When subtracting exponents, you**might get a zero or a negative exponent. Thus, remember to use the Zero - Exponential Rule and the Negative Exponential Rule: x0 = 1, If x = 0**Divide coefficients**Example 1: Divide terms with the same bases = - 5 xy 4 xhasdegree 1; 1 can be omitted**Dividing a Polynomial by a Monomial**If a polynomial of more terms is divided by a monomial, divide each term of the polynomial by the monomial. Remember!Always think about which term is positive or negative**For example:**Remember to divide 3y into every term = y4 + 4y3 – 6y**Dividing a Polynomial by a Binomial**Example: (x3 + 4x2 + 2x – 6) (x + 2) Divisor Dividend We are to find the Quotient to which, by Division fact, when it multiplies the divisor, the product must be equal to the dividend.**The solution to this question can be found by Long Division,**and it usually written in the form: Dividend Divisor Remainder Divisor = Quotient + Long Division can be checked by the following fact: Dividend = Quotient Divisor + Remainder**Long Division Algorithm**When the divisor has more than one term, we will need to find the Quotient by using Long Division. Here is the process of the Algorithm of Long Division:**Arrange all the terms of both dividend**• and divisor from the highest degree to the lowest degree. Set up division as example shown: (4x2 + 2x – 5) (x + 2) Then set up the problem as follows: ) x + 2 4x2 + 2x – 5**Find the first term of the Quotient by**• dividing the first term of the dividend • by the first term of the divisor, as shown. Divide 4x2 by x = 4x 4x ) x + 2 + 4x2 + 2x – 5**Multiply every term of the divisor by**• the first term of the quotient, 4x, and • write the result as shown. 4x ) x + 2 4x2 + 2x – 5 4x2 + 8x Multiply 4x(x+2)**Subtract the product from the dividend**• by changing the signs of every term • of the product to opposite, then add • like terms. 4x ) x + 2 4x2 + 2x – 5 4x2 + 8x Change the sign of each term to its opposite - 6x Combine like terms**Bring down the next term, write it next**• to the first remainder as shown: 4x ) x + 2 4x2 + 2x – 5 Bring down The next term 4x2 + 8x – 6x – 5**Use this last expression as the dividend,**• then repeat the process from step 1 until • the remainder can not be divided any • further. 4x – 6 ) x + 2 4x2 + 2x – 5 4x2 + 8x -- 6x – 6x – 5 Divide - 6xby x = -6 , write –6 next to the first term of the quotient, 4x.**The entire process of this long division**is now shown as follows. quotient 4x – 6 divisor ) x + 2 4x2 + 2x – 5 4x2 + 8x 4x(x+2) – 6x – 5 -6 (x + 2) – 6x – 12 + + 7 remainder**We can express the result of this long**Division as follows:**= 4x2 + 8x – 6x – 12 + 7**and check by showing that the product of the quotient and the divisor, plus the remainder is equal to the dividend. divisor remainder Quotient (4x – 6)(x + 2) + (7) dividend = 4x2 + 2x – 5